Square-free integer

A square-free integer is a number which is not divisible by any square numbers other than 1. In other words, each prime number that appears in its prime factorization appears exactly once.

For example,  is square-free. However,  is not square-free: it is divisible by , and the power of  in the prime factorization is to a power larger than one.

 Möbius function 

The Möbius function is a function which takes in natural numbers and is usually written as . The value of  depends on whether or not  is square-free. Specifically,

For example,  is square-free with an 2 prime factors, so . Since  is not square-free, then .

Since  is prime, it is its own prime decomposition. That is, the prime factorization of  has 1 prime factor, so .

 References 

Category:Number theory